I have never heard any actual expert say any such thing.

Perhaps some non-experts were spouting such nonsense, but nobody should be putting any real importance on what they say.

You're right. They said that vaccines are safe and effective, and they are. Nothing in this world is perfectly safe and effective, and no one qualified to speak on any healthcare option would ever claim otherwise.

Tap water is safe, but non-zero people get sick from it. That just means it's not completely safe.

Penicillin is effective, but some germs are resistant now. That just means it's not completely effective.

There are certainly situations where specific vaccine formations have caused measureable harm. For example in Japan there was a specific formulation of MMR vaccine that led to high rates of sepsis. Experts certainly were involved in the messaging around the changes that Japan applied to resolve this issue (the results also helped provide evidence that vaccines don't cause autism).

Public messaging around human health, especially with respect to powerful tools that modify the immune system, has always been extremely challenging. Public health officials typically want to maximize health for a whole population, and we know that messaging can lead to society-level changes in behavior around vaccination. So experts are typically quite careful when they talk about harms from vaccines.